We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity<gamma</=2). By considering the long-distance asymptotic behavior of these solutions, a connection is esta...

In this paper‎, ‎we have shown that the coset diagrams for the‎ ‎action of a linear-fractional group $Gamma$ generated by the linear-fractional‎ ‎transformations $r:zrightarrow frac{z-1}{z}$ and $s:zrightarrow frac{-1}{2(z+1)}$ on‎ ‎the rational projective line is connected and transitive‎. ‎By using coset diagrams‎, ‎we have shown that $r^{3}=s^{4}=1$ are defining relations for $Gamma$‎. ‎Furt...

We investigate qualitative properties of weak solutions the Dirichlet problem for equation $$-\Delta _p u = \lambda \,m(x)|u|^{p-2}u+ \eta \,a(x)|u|^{q-2}u+ f(x)$$ in a bounded domain $$\Omega \subset \mathbb {R}^N$$ , where $$q<p$$ . Under certain regularity and assumptions on weights m, source function f, we identify ranges parameters $$\lambda $$ $$\eta which satisfy maximum antimaximum prin...

We give a short new proof of large N duality between the Chern-Simons invariants of the 3-sphere and the Gromov-Witten/Donaldson-Thomas invariants of the resolved conifold. Our strategy applies to more general situations, and it is to interpret the Gromov-Witten, the DonaldsonThomas, and the Chern-Simons invariants as different characterizations of the same holomorphic function. For the resolve...

We revisit the problem of small Bjorken-$x$ evolution gluon and flavor-singlet quark helicity distributions in shock wave ($s$-channel) formalism. Earlier works on subject same framework resulted an equation for field-strength $F^{12}$ "axial current" ${\bar\psi}\gamma^+\gamma^5\psi$ operators (sandwiched between appropriate light-cone Wilson lines) double-logarithmic approximation (DLA: summin...

let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...